Directed Numbers Worked ExamplesIGCSE Maths 0580

On a number line, further left is smaller. For negatives, the larger the digit, the smaller the number.

Compare the negatives first: $-3.5$ is further left than $-3$, which is further left than $-0.5$.

All positives sit to the right of all negatives.

Work out the numerator first, following the order of operations. The power comes before the multiplication.

Now the multiplication. A positive times a negative is negative, and subtracting it adds.

Combine the numerator.

Finally divide. A positive divided by a negative is negative.

(a) Add the five values, keeping track of the signs.

Divide the total by how many values there are.

(b) The range is the largest value minus the smallest. The smallest is $-8$, not $-2$.

Substitute with brackets around each negative value, then deal with the numerator and denominator separately.

Cube first — an odd power of a negative stays negative.

Square the second term.

Numerator: subtracting a positive makes it more negative.

Denominator.

Divide.

Substitute, keeping each negative value inside brackets so the signs survive.

Square first. An even power of a negative is positive.

Then the multiplication — subtracting a negative adds.

Combine.

Write the month as a single calculation: money in is positive, money out is negative.

Work left to right.

Then subtract the withdrawal.

The balance is negative, so the account is still overdrawn — by $3.

Read every bar first. Each gridline is $1^{\circ}\text{C}$, and bars below the zero line are negative.

(a) The highest is $2$ (Thursday) and the lowest is $-9$ (Saturday). Subtract the lowest from the highest.

(b) Work out each change as later day minus earlier day, so that rises are positive and falls are negative.

The largest positive change is from Saturday to Sunday.

(c) Add all seven temperatures, keeping the signs, then divide by $7$.

(a) Start where two of three connected bricks are known. The $-1$ sits on the empty bottom brick and the $6$, so work down by subtracting.

Now complete the second row.

Third row, then the top.

(b) Write the middle row of Pyramid 2 in terms of $x$.

The top brick is their product. Two negatives multiply to a positive.

Check both. If $x = 3$, the middle row is $-6$ and $-3$, and $(-6) \times (-3) = 18$. If $x = -3$, the middle row is $6$ and $3$, and $6 \times 3 = 18$.

Read the six balances from the graph. The gridlines are every $5, and points below the zero line are negative.

(a) Leo worked out $40 - 25 = 15$, treating Tuesday's balance as $+25$. The change is later minus earlier.

So the balance fell by $65. (b) List each day-to-day change in the same way.

The only large increase is from Wednesday to Thursday.

(c) A mean of $0$ over seven days means the seven balances add up to $0$. Add the six known balances.

So Sunday's end-of-day balance must be $+70$ to bring the total to $0$.

The question asks for the payment, not the balance. The payment has to take the balance from $-35$ up to $70$.